As soon as you can construct a vector field with finitely many isolated singularities on a non-compact manifold, you can slide them ll the way to infinity and get a vector field with no singular points. If the Poincaré-Hopf index worked, then the Euler characteristic of all non-compact manifolds would vanish.

On the other hand, you can say more useful things. For example, if you can find in your non-compact manifold $M$ a submanifold $M'\subseteq M$ of the same dimension which is compact and with a boundary, such that the inclusion is, say, a homotopy equivalence, then the Poincaré-Hopf theorem works for vector fields with finitely many singular points inside the interior of $M'$ and pointing outward on $\partial M'$.

As soon as you can construct a vector field with finitely many isolated singularities on a non-compact manifold, you can slide them all the way to infinity and get a vector field with no singular points. If the Poincaré-Hopf index worked, then the Euler characteristic of all non-compact manifolds would vanish.

On the other hand, you can say more useful things. For example, if you can find in your non-compact manifold $M$ a submanifold $M'\subseteq M$ of the same dimension which is compact and with a boundary, such that the inclusion is, say, a homotopy equivalence, then the Poincaré-Hopf theorem works for vector fields with finitely many singular points, all inside the interior of $M'$ and pointing outward on $\partial M'$.

Question: Can one always find such an $M'$ if we start with a non-compact $M$ that has finitely many ends and a vector field with finitely many regular singular points?